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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 332 · 40 questions
TalentJR

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

NameDateTime takenScore ___ / 40
  1. 1.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  2. 2.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 324 (c) 16 (d) 4
  3. 3.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 29 (c) 69 (d) 39
  4. 4.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 121 (c) 11 (d) 145
  5. 5.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  6. 6.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 333 (c) 171 (d) 225
  7. 7.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 5194 (d) 4624
  8. 8.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  9. 9.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  10. 10.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 3 (c) 121 (d) 65
  11. 11.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 2 and 15 (d) 5 and 6
  12. 12.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 170 (c) 247 (d) 478
  13. 13.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1444 (c) 1764 (d) 1774
  14. 14.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
  15. 15.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 225 (c) 635 (d) 425
  16. 16.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 14 (d) 1 and 56
  17. 17.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 12 (d) 35
  18. 18.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4499 (c) 4891 (d) 4489
  19. 19.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10619 (c) 10609 (d) 9409
  20. 20.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 12 (c) 11 (d) 28
  21. 21.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 794 (d) 784
  22. 22.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 16 (c) 196 (d) 106
  23. 23.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 20 (d) 19
  24. 24.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 144 (c) 184 (d) 104
  25. 25.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6084 (c) 6734 (d) 6724
  26. 26.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 441 (d) 225
  27. 27.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 11 (c) 1 and 35 (d) 2 and 10
  28. 28.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 58 (c) 100 (d) 4
  29. 29.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 9 (c) 45 (d) 81
  30. 30.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 4 and 5 (c) 2 and 10 (d) 1 and 8
  31. 31.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 25 (c) 1 (d) 13
  32. 32.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
  33. 33.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 851 (d) 841
  34. 34.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  35. 35.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 8 (c) 18 (d) 10
  36. 36.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
  37. 37.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 256 (d) 136
  38. 38.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 904 (d) 1024
  40. 40.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 14 (d) 56

Answer key

  1. (x + 8)(x + 9)
  2. 16
  3. 29
  4. 121
  5. (x + 3)(x + 6)
  6. 117
  7. 5184
  8. (x + 6)(x + 9)
  9. (x + 7)(x + 8)
  10. 9
  11. 5 and 6
  12. 170
  13. 1764
  14. (x + 2)(x + 7)
  15. 625
  16. 7 and 8
  17. 12
  18. 4489
  19. 10609
  20. 11
  21. 784
  22. 16
  23. 20
  24. 104
  25. 6724
  26. 225
  27. 5 and 7
  28. 16
  29. 9
  30. 4 and 5
  31. 1
  32. (x + 2)(x + 3)
  33. 841
  34. (x + 7)(x + 9)
  35. 9
  36. 113
  37. 136
  38. (x + 3)(x + 4)
  39. 1024
  40. 15

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